General Topology/Continuity

Continuity is a local property, in that it may be characterized by a property that a function might have at every point.

When $$Y$$ is a uniform space, the definition of equicontinuity simplifies, and furthermore in this situation equicontinuous subsets are related to compact subsets of $$\mathcal C(X,Y)$$. This we will see in the chapter on uniform structures.

In other words, a function $$f: X \to Y$$ is a local homeomorphism if and only if for all $$x_0 \in X$$, there exists an open neighbourhood $$U$$ of $$x_0$$ and an open neighbourhood $$V$$ of $$f(x_0)$$ so that $$f|_U$$ is a homeomorphism from $$U$$ to $$V$$.